


Abstract
F. Schieweck, Sat, 10.0010.25

Previous

Contents

Next


  

A new Stokes solver based on Lagrange multipliers
F. Schieweck (OttovonGuericke Universität Magdeburg)
We consider the discrete Stokes problem generated by a finite
element method with continuous Q_{r}elements for the velocity
components and discontinuous P_{r1}elements
for the pressure. Our aim is to construct an efficient solver for
the arising indefinite system of equations. The idea is to solve
the problem for a larger velocity space, which contains also
discontinuous velocity modes, and to handle the desired continuity of
the velocity by means of Lagrange multipliers. This allows a
decoupling into elementbyelement calculations.
Another key ingredient of our method is the orthogonal
decomposition of the larger velocity space into a discrete
divergencefree subspace of the original conforming velocity
space and a discontinuous complement space.
Thus, the solution of the Stokes problem splits into two
symmetric positive definite (s.p.d.) subproblems. The first one is
an s.p.d. problem for the discrete divergencefree part of the
velocity where the pressure is eliminated. This problem can be solved
efficiently by a multigrid method. The second subproblem is an s.p.d.
problem for the Lagrange multipliers. It can be shown that the
condition number of the corresponding operator is O(1) for h > 0.
Therefore, a CGmethod is an efficient solver.
The ideas can be applied also to the NavierStokes equations.




Previous

Contents

Next


